Phase separation in the binary-alloy problem: The one-dimensional spinless Falicov-Kimball model.

The ground states of the one-dimensional Falicov-Kimball model are investigated in the small-coupling limit, using nearly degenerate perturbation theory. For rational electron and ion densities, respectively, equal to p/q , pi /q , with p relatively prime to q and pi /q close enough to 1 2, we find that in the ground state the ion configuration has a period q . The situation is analogous to the Peierls instability, where the usual arguments predict a period-q state that produces a gap at the Fermi level and is insulating. However for pi/q far enough from 1 2, this phase becomes unstable against phase separation. The ground state is a mixture of a period-q ionic configuration and an empty ~or full! configuration, where both configurations have the same electron density to leading order. Combining these results with those previously obtained for strong coupling, it follows that a phase transition occurs in the ground state, as a function of the coupling, for ion densities far enough from 1 2 . @S0163-1829~96!00924-1#